**Cyclotron maser radiation from inhomogeneous plasmas**

R. A. Cairns, I. Vorgul, R. Bingham, K. Ronald, D. C. Speirs, S. L. McConville, K. M. Gillespie, R. Bryson, A. D.

R. Phelps, B. J. Kellett, A. W. Cross, C. W. Roberston, C. G. Whyte, and W. He

Citation: Physics of Plasmas (1994-present) 18, 022902 (2011); doi: 10.1063/1.3551697

View online: http://dx.doi.org/10.1063/1.3551697

View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/18/2?ver=pdfcov

Published by the AIP Publishing

R. A. Cairns,1I. Vorgul,1R. Bingham,2,3K. Ronald,3D. C. Speirs,3S. L. McConville,3 K. M. Gillespie,3R. Bryson,3 A. D. R. Phelps,3 B. J. Kellett,2 A. W. Cross,3 C. W. Roberston,3C. G. Whyte,3and W. He3

1

*School of Mathematics and Statistics, University of St. Andrews, St. Andrews,*
*Fife KY16 9SS, United Kingdom*

2

*STFC Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, United Kingdom*

3

*Department of Physics, University of Strathclyde, Glasgow G4 0NG, United Kingdom*

共Received 20 July 2010; accepted 6 January 2011; published online 16 February 2011兲

Cyclotron maser instabilities are important in space, astrophysical, and laboratory plasmas. While extensive work has been done on these instabilities, most of it deals with homogeneous plasmas with uniform magnetic fields while in practice, of course, the systems are generally inhomogeneous. Here we expand on our previous work关R. A. Cairns, I. Vorgul, and R. Bingham, Phys. Rev. Lett.

**101**, 215003 共2008兲兴 in which we showed that localized regions of instability can exist in an
inhomogeneous plasma and that the way in which waves propagate away from this region is not
necessarily obvious from the homogeneous plasma dispersion relation. While we consider only a
simple ring distribution in velocity space, because of its tractability, the ideas may point toward
understanding the behavior in the presence of more realistic distributions. The main object of the
present work is to move away from consideration of the local dispersion relation and show how
global growing eigenmodes can be constructed. ©*2011 American Institute of Physics*.

关doi:10.1063/1.3551697兴

**I. INTRODUCTION**

Cyclotron maser instability1 is important in the labora-tory for the generation of high power radiation in devices like gyrotrons and is also thought to play a crucial role in the generation of auroral kilometric radiation共AKR兲and similar electromagnetic wave emissions from other astrophysical ob-jects including planets,2–6 stars,7–10 brown dwarfs,11and as-trophysical shocks.12In previous work we have analyzed the instability produced by a horseshoe shaped distribution in electron velocity space and suggested that it is a likely source of AKR.2,7,13A laboratory experiment and simulations in which this type of distribution was produced showed that it did indeed generate waves with many of the properties of AKR.14–19 In the simulations and in the laboratory experi-ment an electron beam with initial finite pitch spread

␣=*v*_{⬜}/*vz* was injected into an increasing axial magnetic

field. Through conservation of magnetic moment, a progres-sive magnetic mirroring of the beam resulted in the conver-sion of axial momentum into perpendicular momentum yielding a horseshoe shaped velocity distribution. Such a process has been observed5 to occur in the AKR source re-gion. The electrons are accelerated into the increasing mag-netic field of the terrestrial auroral magnetosphere.20The ef-ficiency of conversion of electron energy into rf energy in both the experiment and simulations is typically 1%, which compares well with the observed natural AKR generation efficiency. The experimental work and simulations have strong relevance to the Interrelationship between Plasma Ex-periments in Laboratory and Space共IPELS兲community, with results having been presented at the associated biennial con-ference and published in an IPELS special issue of Plasma

Physics and Controlled Fusion共PPCF兲.14,18Both theory and experiment indicate that the radiation is generated in the ex-traordinary共X兲mode below the local cyclotron frequency. If we look at the cold plasma dispersion curves for the X mode perpendicular or near perpendicular to the magnetic field, then the branch which propagates below the cyclotron fre-quency does not connect to the branch which propagates in the vacuum. However, it is clear that the AKR does escape the region where it is generated since it is observed outside planetary magnetospheres. This is a problem which has at-tracted a number of suggested solutions over the years,6but no definitive answer. In a previous paper,21we considered a plasma with a particularly simple and analytically tractable distribution, a ring in velocity space, and showed that in a slab geometry in which this unstable distribution was con-fined to a local region, there are unstable eigenmodes and that radiation then propagates away in both directions from the slab, regardless of whether the field outside the slab is greater or less than that inside. We also gave some prelimi-nary analysis of the behavior in a smoothly varying inhomo-geneity. The purpose of the present paper is to expand on this work and show that with a suitable magnetic field configu-ration it is possible for the wave to grow in a localized re-gion. The basic result underlying our work is that if a suit-able complex frequency is chosen, with positive growth rate, then at a localized point in a magnetic field gradient forward and backward waves couple and there is partial reflection. If two such points exist in the magnetic field profile then a standing wave can be set up in the region between them. This allows the existence of temporally growing eigenmodes of the system.

1070-664X/2011/18共2兲/022902/5/$30.00 **18**, 022902-1 © 2011 American Institute of Physics

**II. THEORY**

We look at an electron distribution of the simple form

*f*共*p*储,*p*⬜兲=共*n*0/2*p*⬜兲␦共*p*储兲␦共*p*⬜−*p*0兲 in momentum space.

The dispersion relation for this can be found by substituting it into the standard expression for the dielectric tensor in a plasma, including relativistic effects which are important. For propagation perpendicular to a steady magnetic field we get the relation

*k*2=2

## 冉

储−_{⬜}2

储

## 冊

, 共1兲

with

储= 1 −

1 2␥

*pe*
2

共−⍀兲+

*T*

4␥2

*pe*2

共−⍀兲2,

⬜= 1 −储,

␥=

## 冉

1 +*p*0

2

*m _{e}*2

*c*2

## 冊

1/2,

*T*= *p*0

2

*me*
2

*c*2,

⍀=*ce*

␥ ,

with*ce*=*eB*/*me*, the electron cyclotron frequency. In

deriv-ing this it has been assumed that the wave frequency is close
to*ce*so that only one of the infinite series of terms in the

dispersion relation is important and that the wavelength is
much greater than the electron Larmor radius, so that in the
Bessel functions which appear we can make the
approxima-tion*J*1共*z*兲⬇*z*/2. Typical dispersion curves arising from this

relation are shown in Fig. 1 for ⍀= 10 and ␥= 1.02. The unstable modes correspond to the almost horizontal branches of the real part. This is clearly a very simple distribution unlikely to occur in naturally occurring plasmas. Strangeway22 carried out a detailed study of the effects of thermal spread and a background plasma on the ring distri-bution. However, our objective here is to show how to go beyond a consideration of the local dispersion relation and demonstrate the existence of growing eigenmodes in an inhomogeneous system. To develop the methodology for this it is convenient to start with this analytically tractable distribution.

Now we consider a nonuniform plasma with the
mag-netic field as a function of*x*, the coordinate along*k*
perpen-dicular to the magnetic field. From Eq.共1兲it is clear that the
local value of *k*2 _{is uniquely defined at each value of} _{x}

共which appears in the dispersion relation through an*x*
depen-dence of⍀兲 and that if is real then so is*k*2_{. This means}

that the wave either propagates without growth or damping
or is evanescent if*k*2_{is negative. In the neighborhood of the}

cyclotron resonance,*k*2 _{is positive and varies smoothly with}

*x*, as shown in Fig.2, so that a wave with realpropagates
through this region and the instability does not go over into a
spatial amplification.

If, however, we givean imaginary part comparable to
the growth rate found in the homogeneous case, then we get
the behavior shown in Fig.3, where*k*comes close to passing
through zero and varies rapidly in the region of cyclotron
resonance. The result of this is that the left and right
propa-gating waves no longer pass through the resonance region

5 10 15 20

5 10 15 20

k

5 10 15 20

0 0.05 0.1 0.15

k

[image:3.612.320.559.51.247.2]Re( )ω _{Im}_{( )}_{ω}

FIG. 1.共Color online兲Real and imaginary parts of共scaled to*pe兲*as a

function of*k*共scaled to*pe*/*c*兲for␥= 1.02.

10

− −5 0 5 10

9 10 11 12 13 14 15

x

k(

x

)

FIG. 2. 共Color online兲The wavenumber*k*with= 10 and⍀= 10− 0.05*x*.

10

− −5 0 5 10

5

−

0 5 10 15

x

k

(x

[image:3.612.52.299.52.132.2])

FIG. 3. 共Color online兲Real共full兲and imaginary共dotted兲parts of*k* with

= 10+ 0.13*i*and⍀= 10− 0.05*x*.

022902-2 Cairns*et al.* Phys. Plasmas**18**, 022902共2011兲

independently, but interact so that a purely outgoing wave on one side connects to a mixture of incoming and outgoing waves on the other side. Previously we suggested that it is possible, for a suitable choice of the growth rate, for a mode to grow locally around the cyclotron resonance and to pro-duce outward propagating waves on both sides, as needed for a physically realistic solution. However, as we shall show, when there is just one position of cyclotron resonance there is always an incoming wave on one side, although its ampli-tude may be small until we get to some considerable distance from the resonance. What is needed for a solution which really does have only outgoing waves at infinity is a field configuration in which the wave passes through resonance twice.

We first elaborate on what happens with a simple
lin-early varying field with⍀= 10− 0.05*x*, as in the plots above.
If we choose the value ofcorrectly then we get the sort of
behavior shown in Fig. 3, but with the real and imaginary
parts of *k* passing through zero together 共which is not the
case in Fig.3兲. A value for which this happens, at least to a

very good approximation, is 10+ 0.1298*i*. In finding this the
real part has been chosen to be 10, but with a different value
of the real part the roots come together at a different point in
the magnetic field profile, with a different imaginary part.
Corresponding to the dispersion relation 共1兲, we can

con-struct a differential equation for the wave electric field which takes the form

*d*2*E*
*dx*2 = −*k*

2_{共}_{x}_{兲}_{E}_{,} _{共}_{2}_{兲}

and whose solution with the boundary condition of an out-going wave on the left共i.e., the high field side兲is as shown in Fig.4.

Far from the resonance region around*x*= 0 the solution
of the dispersion relation is close to *k*= 共in scaled units兲
and so there is straightforward exponential growth or decay
with the wave amplitude decreasing along the direction of
propagation. This is not damping, just a consequence of the
fact that the wave source is growing in time. So, in Fig.4we
see an outgoing wave on the left, while on the right the
solution for small*x*is dominated by a wave propagating to

the left, but eventually an incoming wave becomes evident
and dominates for large enough*x*. Previously, we attributed
this incoming wave to not having found exactly the correct
value ofor to numerical error, but now we will argue that
it would be expected to appear.

If we look at the neighborhood of the zero of *k*2 _{then}

locally we can consider its variation with *x* to be linear,
although it does not take real values. Nevertheless, the local
behavior is described by Airy’s equation, just as for a cut-off
in which*k*2just goes from positive to negative. Regardless of
the complication of having a complex coefficient in Airy’s
equation, the way in which the solution connects through
zero is the same, in that a solution going as either
exp关⫾*i*兰*k*共*x*兲*dx*兴 on one side connects to a superposition of
equal amplitudes of the two waves on the other side. If we
look at the behavior, which is close to that shown in Fig.3,
we see that far to the left *k* is approximately constant and,
having set up an outgoing wave on this side we get the
amplitude increasing exponentially with *x*. Near *x*= 0 the
imaginary part of*k*becomes large and there is a very rapid
growth of the amplitude. On the other side of the resonance,
the outgoing wave, which corresponds to the positive Re共*k*兲

root plotted in the diagram, initially has Im共*k*兲negative and
continues to grow until Im共*k*兲changes sign and the amplitude
falls. However, as we pass through*k*= 0 we expect to pick up
an equal amplitude incoming wave. Initially this has Im共*k*兲

large and positive and falls to a very small level until Im共*k*兲

changes sign and it starts to grow, eventually becoming
com-parable in amplitude to the outgoing wave and producing the
oscillations around*x*= 20 then becoming dominant for large

*x*. To check this interpretation we have looked at a WKB
approximation starting with the assumption of equal
ampli-tudes at the*k*= 0 point which is at *x*0= 0.975. In Fig. 5 we

show

*I*共*x*兲=

### 冕

*x*_{0}
*x*

Im关*k*共*y*兲兴*dy*.

This takes the value zero for*x*a little above 20, so that at
this point the incoming and outgoing waves have the same
60

− −40 −20 0 20 40

0.1 10 1 10× 3 1 10× 5 1 10× 7

1 10× 9 1 10×

x

Mo

d

u

lus

o

f

[image:4.612.316.559.48.244.2]E

FIG. 4.共Color online兲 兩*E*兩as a function of*x*from the differential equation.

0 10 20 30

6

−

4

−

2

−

0

I x( )

x

FIG. 5. 共Color online兲*I*共*x*兲as a function of*x*.

amplitude. It is clear that this agrees with the behavior found numerically. The connection of an outgoing wave on one side to incoming and outgoing waves of comparable ampli-tude on the other side of the resonance does not depend on the wavenumber passing through zero exactly. The sort of behavior shown in Fig.3where the real part comes close to zero or, sometimes, dips below zero is sufficient.

Since a solution with only outgoing waves at infinity is
impossible with a simple linear field gradient, we now turn to
the question of when it is possible. The answer, clearly, is to
have a profile with two resonance positions such that there
are left and right propagating waves between the two and
they match to purely outgoing waves on the outside. This
requires that not only the amplitude but the phase matches up
correctly, giving rise to the sort of eigenvalue problem to be
expected for a system of this sort. We will look at a situation
where there are many wavelengths between the resonance
points and the eigenvalues are consequently closely spaced.
Rather than look at the detailed structure of these
eigenval-ues we will simply look for an approximate matching of the
amplitude, a much simpler problem which should give the
overall trend of the eigenvalues. Specifically we look at a
profile with ⍀共*x*兲= 9 + 0.5*x*2_{. With waves of frequency of}

around 10 that we have looked at, there will then be two
resonances across which we can expect outgoing waves in
the high field region outside to connect to waves propagating
in both directions in the central region where the field is
lower. In Fig. 6 we show a numerical solution with = 11
+ 0.0995*i*. While this is evidently not an exact eigenvalue of
the problem, it is clear that the amplitudes of the incoming
and outgoing waves come very close to matching at the
cen-ter so that we are not far away from an eigenvalue. The
behavior of the solution is very sensitive to the choice of,
so that this will give a good approximation to an eigenvalue.
In Fig. 7 we show the trend of the eigenvalues for a
number of values estimated in this way. As pointed out
be-fore, the exact values would be expected to be a closely
spaced series following this line. The growth rates found are,
as might be expected, a little below the maximum value
found for a homogeneous plasma. The lowest real frequency

we have found is 9.2 and it does not seem possible to find solutions with a value much below this. Similar behavior is found for parallel propagation, although in this case the be-havior across the resonance is the reverse of that found for perpendicular incidence. The result is that instability occurs in regions of high field with small amplitude waves propa-gating away into the surrounding low field region.

If we go back to our consideration of a linear magnetic
field gradient we can note that there may be other possible
ways for a localized instability to occur. From Fig.6it can
be seen that for the parameters presented there the ratio of
the outgoing wave amplitude to the incoming wave
ampli-tude reaches a maximum value of around *e*5, which
corre-sponds to an intensity ratio in excess of 104. So, if there was
some sort of discontinuity in the plasma parameters which
produced a reflection of more than 10−4 _{of the incident }

en-ergy flux, perhaps a steep density jump, then this would, again, open up the possibility of having a resonant cavity with right and left traveling waves between the cyclotron resonance and the reflection point connecting to outgoing waves on each side.

**III. CONCLUSION**

We have analyzed a simple cyclotron maser instability in an inhomogeneous plasma and have shown that it has prop-erties which are not obvious from the dispersion relation for a homogeneous system. Convective growth of a wave with real frequency passing through a cyclotron resonance does not seem to occur. On the other hand, if there is a suitable magnetic field profile a temporally growing wave can be reflected between two regions of cyclotron resonance, with some of the energy leaking away into the surrounding plasma. The possibility of obtaining instability with only one resonance region and some sort of reflecting layer in the plasma is also pointed out. The ring distribution has been chosen because of its analytically tractable nature, the main point of the present analysis being to show how to go beyond a consideration of the local dispersion relation and show that global growing eigenmodes exist in a suitable

inhomoge-20

− −10 0 10 20

10 100 1 10× 3 1 10× 4

x

Mo

d

u

lus

o

f

[image:5.612.327.556.49.244.2]E

FIG. 6. 共Color online兲Numerical solution for= 11+ 0.0995*i* with an

in-coming wave to the right and an outgoing wave to the left.

9 10 11 12

0.09 0.1 0.11 0.12 0.13 0.14 0.15

Real part

Im

ag

ina

ry

p

ar

[image:5.612.51.302.52.231.2]t

FIG. 7. 共Color online兲Real and imaginary parts of complex eigenvalues.

022902-4 Cairns*et al.* Phys. Plasmas**18**, 022902共2011兲

analysis with more realistic distributions of more direct rel-evance to problems in space and astrophysics.

**ACKNOWLEDGMENTS**

R.B. would like to thank the STFC Centre for Funda-mental Physics for support. This work was supported by EPSRC Grant No. EP/G04239X.

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